2016 American Control Conference (ACC) Boston Marriott Copley Place July 6-8, 2016. Boston, MA, USA Structural Analysis and Design of Dynamic-Flow Networks: Implications in the Brain Dynamics

Sergio´ Pequito † Ankit N. Khambhati \,[ George J. Pappas † Dragoslav D. Siljakˇ ‡ Danielle Bassett †,\,[ Brian Litt \,[,]

Abstract— In this paper, we study dynamic-flow networks, the edges are assumed constant over time. However, this is a i.e., networks described by a graph whose weights evolve rather rigid description that often fails to adequately represent according to linear differential equations. Further, these linear the structure of natural, social and man-made processes, differential equations depend on the incidence relation of the edges in a node, and possibly nodal dynamics. Because some especially in uncertain environments where the relations of these weights and their dependencies may not be accurately between the constituent parts change over time [2]. known, we extend the notion of structural controllability for Dynamic-flow networks model dynamical systems cap- dynamic-flow networks, and provide necessary and sufficient tured by networks where the weights on the edges evolve conditions for this to hold. Next, we show that the analysis according to linear differential equations. In addition, these of structural controllability in dynamic-flow networks can be reduced to that of a digraph which we refer to as meta digraph. linear differential equations depend on the incidence relation In addition, we consider different actuation capabilities, i.e., we of the edges in a node, and possibly nodal dynamics. For ex- assume that both the nodes and edges in the dynamic-flow ample, in social networks, a node (representing an individual) network can be actuated, and we explore the implications constantly processes information received from its upstream in terms of computational complexity when the minimum neighbors and makes decisions that are communicated to its cost-placement of actuators is considered. The proposed framework can be used to identify actua- downstream neighbors. The information received and passed tion capabilities required to mitigate epileptic-brain dynamics. by a node can be represented by the state variables on its More precisely, the functional connectivity of mesoscale brain incoming and outgoing edges. Thus, mapping the signals of dynamics can be modeled as a dynamic-flow network by the incoming edges onto those of the outgoing edges. This considering dynamic functional connectivity of the network. is also the case in a network of computers and routers on In the context of epilepsy, the modeling is motivated by new findings that show that the edges within seizure-generating the Internet, where the edges represent physical connections areas are almost constant over time, whereas the edges outside and the state variables on the edges represent the amount these areas exhibit higher variability over time in human epilep- of packet flow along a particular connection in a given tic networks. In addition, implementable devices to control direction. The mechanism of the nodes then corresponds to drug-resistant seizures by affecting the epileptic network has a load-balancing or routing mechanism that allows packets gained considerable attention as a viable treatment option. Subsequently, from a control-theoretic perspective, one can to reach their destination while avoiding congestion. consider actuation to attenuate edge variability responsible Dynamic-flow networks can also be used to model func- for seizure-generation in the epileptic network. In particular, tional brain networks, where nodes represent neural popula- we address the following two scenarios: (i) current placement tions and edges are statistical relationships between neural of electrical stimulators, and their probable capabilities; and activation patterns. In particular, these networks are suitable (ii) determine the minimum cost placement with minimum actuation capabilities. The latter problem is motivated by the to capture the brain dynamics when dynamic functional fact that some edges may correspond to more accessible (or less connectivity is considered [3]. More specifically, a sliding- harmful) regions in the brain, whereas others might correspond window over the blood-oxygen level dependent (BOLD) to sensitive regions in the brain. signal is considered to obtain over time the functional connectivity that captures the Pearson’s correlation between I.INTRODUCTION signals. In particular, the local dependency on adjacent edges Complex networks have traditionally been studied by is justified by the spatial-temporal dependency of the BOLD resorting to graphs [1]. These have been considered most signals [3]. of the times as static objects, whose weights associated with The control of dynamic-flow networks was introduced in [4], and in this paper we aim to account for the case This work was supported in part by the TerraSwarm Research Center, one of six centers supported by the STARnet phase of the Focus Center where the parameters describing the possible interaction Research Program (FCRP) a Semiconductor Research Corporation program dependencies in the flow-dynamic networks are either free or sponsored by MARCO and DARPA, and the NSF ECCS-1306128 grant. constant parameters over time. Towards this goal we resort ‡Department of Electrical Engineering, Santa Clara University, Santa Clara, CA 95 053, USA to structural systems theory [5] that enables the introduction †Department of Electrical and Systems Engineering, School of Engineer- of generic controllability notions [6], i.e., controllability ing and Applied Science, University of Pennsylvania holds for almost all possible parameterization of the free \ Department of Bioengineering, University of Pennsylvania parameters. Therefore, we aim to assess when flow-dynamic [ Penn Center for Neuroengineering and Therapeutics, University of Pennsylvania networks (seen as a dynamical system) are generically con- ] Department of Neurology, Hospital of the University of Pennsylvania trollable. In addition, we assume that different functions and

978-1-4673-8681-4/$31.00 ©2016 AACC 5758 locations of control inputs are possible, which influences the can no longer be done resorting to dual graphs, see Remark 1 edges’ weights and nodes’ dynamics. More precisely, these for additional details. ◦ can be multi-node input, i.e., the input signal is injected in The main contributions of this paper are as follows: (i) we several nodes to regulate how the dynamics of the outgoing formally introduce the notion of generic controllability for edges changes, and multi-edge inputs, i.e., an input can dynamic-flow networks, where some edges’ weights can be actuate a linear combination of edges, not necessarily the constant over time; (ii) we introduce necessary and sufficient ones that share vertices. Particular cases of these two are the conditions that ensure structural controllability; (iii) we ad- out-node inputs, i.e., all the outgoing edges from a single dress the problem of minimum actuator placement incurring vertex are driven by a scalar input signal, and the in-edge in the minimum cost while ensuring structural controllability; inputs, i.e., an input can actuate only a single edge dynamics (iv) we show how dynamic-flow networks can identify brain directly. In this paper, we analyze and provide necessary regions that should be actuated to ensure that edges’ weights and sufficient conditions for dynamic-flow networks, where variations are kept within certain bounds; (v) we show that parameters can be either free or constant over time, to current actuation capabilities are enough to ensure structural be generically controllable. In addition, we consider the controllability; and (vi) using real data, we determine the control placement problems under different assumptions; location of actuators to regulate the dynamics such that more precisely, we consider that different edges and/or nodes epileptic seizures may be attenuated and/or overcome. may incur in different costs. The rest of the paper is organized as follows. In Sec- tion II, we provide the formal problem statement. Section III Related Work reviews some concepts from computational complexity and Several necessary and sufficient conditions that character- graph theoretical concepts used in structural systems theory. ize the structural controllability, as well as their verification, Section IV presents the main technical results, followed are known for linear time-invariant [5], or switching sys- by a case study in Section V showing the implications tems [7]. Nevertheless, the selection, commonly referred to in the brain dynamics; more specifically, in the context of as design, of minimum actuation capabilities to ensure that epilepsy. Conclusions and discussions on further research are these conditions hold has only been addressed in the last presented in Section VI. years. In the context of linear time-invariant systems, the problem of determining the minimum number of actuated II.PROBLEM STATEMENT variables ensuring structural controllability was addressed in [8], [9]. Later, it was extended to the case where the mini- Let D = (V, E) be a digraph, where V denotes a set of n mum cost is sought when the actuated state variables incur in vertices, and E the set of m directed edges been the vertices. In addition, consider that each edge in the digraph has weight a cost that does not depend on the actuator considered [10], + and to the minimum cost problem allowing the actuation cost w associated with it, where w : E → R0 , and a label µ(e) ∈ of a state variable to depend on the actuator considerez [11]. {1,?}, for e ∈ E, where µ(e) = 1 indicates that the weight An alternative formulation consists in assuming a predefined is constant over time and µ(e) = ? indicates that it varies collection of actuators from which one selects those to be over time. Further, let N = {Ni}i∈I represent the multi- used to ensure structural controllability, but in such scenario node actuation signal, where the collection of the nodes’ set the problem of determining the minimum collection of such Ni ⊂ {1, . . . , n} is actuated by the actuator i. In particular, actuators is NP-hard [12]. Notwithstanding, some subclasses if every Ni contains only one node than we have out-node have been determined where the problem is polynomial, even inputs. Similarly, let E = {Ej}j∈J be the collection of the when different actuation costs are considered [13]. More edges’ set Ej ⊂ E actuated by the actuator j, and if every recently, actuation selection problems ensuring structural Ej contains only one edge than we have in-edge inputs. controllability were also proposed for discrete-time fractional Consequently, the dynamic-flow network can be described by dynamics [14], and switching systems for some switching (D = (V, E), w, µ, E, N), where it is implicitly assumed that policy [15] and for all switching policies [16]. the dynamics of one edge depends on a linear combination In [17] is presented a framework that is close to ours, on the incoming edges’ weights. A dynamic-flow network where switch-board dynamics is studied for the edges’ dy- is structurally controllable, if there exits control signals namics and assumed to be controlled by in-node inputs, i.e., delivered to the dynamic-flow network by (E, N) such that an external input is injected in the node from which the edges the edges’ weights can be steered to an arbitrary value, for to be actuated depart. In addition, all the edges are actuated almost all free parameters describing edge dynamics. in equal manner, and the dynamics of a node is considered Therefore, we are interested in addressing the following to be constant in time, which we refer to as static node. The problems: analysis results in studying the line/dual graph, on which P1 When is the dynamic-flow network (D, w, µ, E = some actuation deployment strategies are provided. In this {Ej}j∈J , N = {Ni}i∈I ) structurally controllable? paper, we extend [17] in multiple ways: (i) we assume that P2 Consider a structurally controllable dynamic-flow net- the nodes can have dynamics; (ii) we deal with multi-node work (D = (V, EV,V ), w, µ, E = {Ej}j∈J , N = E N E N and multi-edge inputs; and (iii) some edges’ weights may be {Ni}i∈I , {cj }j∈J , {ci }i∈I ), where cj and ci are constant over time. Furthermore, we notice that the analysis the actuation costs associated with Ej and Ni, respec-

5759 tively. What is the minimum |I| and |J | that ensures referred to as state vertices) and EX ,X = {(xi, xj): Aji 6= the lowest cost and structural controllability? 0} denotes the set of state edges. Similarly, we define the following digraphs: D(A,¯ B¯) = (X ∪U, E ∪E ) where III.PRELIMINARIESANDTERMINOLOGY X ,X U,X U represents the set of input vertices and EU,X = {(ui, xj): We start by reviewing some computational complexity B¯ji 6= 0} the set of input edges. concepts [18], followed by some graph theoretic concepts A digraph Ds = (Vs, Es) with Vs ⊂ V and related with the study of structural systems theory [5], [9]. Es ⊂ E is called a subgraph of D. If Vs = V, A (computational) problem is said to be reducible in Ds is said to span D. A sequence of directed edges polynomial time to another if there exists a procedure to {(v1, v2), (v2, v3), ··· , (vk−1, vk)}, in which all the vertices transform the former to the latter using a number of opera- are distinct, is called an elementary path from v1 to vk, tions which is polynomial in the size of its elements. Such a as well as a vertex in a digraph with no incoming and reduction is useful in determining the complexity class that outgoing edges (with some abuse of terminology). A vertex a problem belongs to [18]. For instance, a problem P in NP with an edge to itself (i.e., a self-loop), or an elementary (i.e., the class of non-deterministic polynomial algorithms) path from v1 to vk comprising an additional edge (vk, v1), is said to be NP-complete if all other NP problems can is called a cycle. Finally, a subgraph with some property P be polynomially reduced to P [18]. The NP-complete class is maximal if there is no other subgraph Ds0 = (Vs0 , Es0 ) is used to describe the complexity of decision versions of of D, such that Ds is a subgraph of Ds0 , and Ds0 satisfies problems. For instance, the following constitutes a decision property P . Subsequently, a digraph D is said to be strongly problem that is particularly relevant in the design of structural connected if there exists an elementary path between any systems context [12]: Given the structural dynamics’ matrix pair of vertices. A strongly connected component (SCC) is a and a collection of possible inputs, is there a collection of maximal subgraph DS = (VS, ES) of D such that for every inputs with exactly k elements, such that the system actuated v, w ∈ VS there exists a path from v to w and from w to v. by these is structurally controllable? Alternatively, it is often natural to consider the optimiza- IV. MAIN RESULTS tion versions associated with the decision problems. For In this section, we present the main results of the present instance, the optimization version of the problem stated paper. First, we show that the dynamic-flow network can above aims to determine the minimum k such that the be written in terms of a state-space representation, which aforementioned property holds. This optimization problem we refer to as meta digraph. In particular, a systematic is referred to as the constrained minimum structural input procedure to construct this digraph is presented in Algo- selection (CMSIS) problem [12]. Note that, if a solution to rithm 1. Then, using the meta digraph, we obtain necessary the optimization problem is known, the decision problem and sufficient conditions to ensure structural controllability is straightforward to solve. Consequently, the optimization of the dynamic-flow network (Theorem 1); hence, providing problem formulations of NP-complete problems, are referred an answer to P1. Subsequently, due to the proposed reduction to as being NP-hard, since they are at least as difficult as from the dynamic-flow network to the meta digraph, and the NP-complete problems; in other words, by solving an the fact that a meta digraph can be an arbitrary digraph instance of the optimization problem (the NP-hard problem), (Theorem 3), we can obtain a series of results that dictate the one can obtain a solution to an NP-complete problem. computational complexity of the problem presented in P2 as Finally, we notice that CMSIS is a NP-hard problem [12]. well as its variations; more precisely, the general problem Consider a linear time-invariant system described by is NP-hard, but a polynomial complexity procedure can be used to determine solutions to a subclass of P . x˙ = Ax + Bu, (1) 2 The polynomial reduction from a dynamic-flow network to where x ∈ Rn denotes the state and u ∈ Rp the input. In the meta digraph is presented in Algorithm 1. This reduction addition, A is the n×n matrix describing the autonomous dy- allow us to provide a characterization of the solutions to P1, namics and B the n × p input matrix describing the actuated as described in the next theorem. state variables from each input. Suppose that the sparsity Theorem 1: The dynamic-flow network (D = (V, EV,V ), E N pattern, i.e., location of zeros and (possibly) nonzeros, of w, µ, E = {Ej}j∈J , N = {Ni}i∈I , {cj }j∈J , {ci }i∈I ), E N A and B is available, but the specific numerical values where cj and ci are the actuation costs associated with Ej of the remaining elements is not known. Subsequently, let and Ni respectively, is structurally controllable if and only A¯ ∈ {0, 1}n×n (respectively B¯ ∈ {0, 1}n×p) is the binary if the meta digraph described by the pair (E,¯ B¯) obtained matrix that represents the structural pattern of A (respectively from Algorithm 1 is structurally controllable. In addition, the B), i.e., it encodes the sparsity pattern of A (respectively B) actuation cost achieved by the deployed actuation capabilities by assigning 0 to each zero entry of A (respectively B) and in either digraph is the same.  1 otherwise. Proof: The proof follows by structural induction; more The following standard terminology and notions from precisely, we show that there are few atomic structures graph theory can be found, for instance, in [9]. Let D(A¯) = that hold and the result follows by noticing that any graph (X , EX ,X ) be the digraph representation of A¯ in (1), where consists in a combination of these atoms. First, consider the the vertex set X represents the set of state variables (also case where one node has several incoming and outgoing

5760 ALGORITHM 1: Determine the meta digraph from the to the dynamics of the outgoing edges in node i, see case state digraph. 4 in Figure 1. Finally, we notice that by specializing the Input: The dynamic-flow network and actuation costs construction to the case where the inputs are considered, it immediately follows that the cost associated to the inputs in (D = (V, EV,V ), w, µ, E = {Ej}j∈J , N = E N the meta digraph is the same as the actuation cost in the {Ni}i∈I , {cj }j∈J , {ci }i∈I ). Output: Meta digraph dynamic-flow network. ∗ ∗ ∗ D (E,¯ B¯) = (E ∪ U , EE∗,E∗ ∪ EU ∗,E∗ ) and actuation cost ck associated with the input uk, for k = 1,..., |I| + |J |. E∗ = ∅; U ∗(E) = ∅; U ∗(N) = ∅; U ∗ = ∅; ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ EE ,E = ∅; EU ,E = ∅; EU (E),E = ∅; EU (N),E = ∅; ∗ 1: E = {e ∈ EV,V : µ(e) = ?}; ∗ 2: U (E) = {uj : j ∈ J } where the cost of input uj is E cj for j ∈ J . ∗ 3: U (N) = {u|J |+i : i ∈ I} where the cost of input N u|J |+i is ci for i ∈ I. 4: U ∗ = U ∗(E) ∪ U ∗(N); 0 ∗ ∗ 0 0 5: EE∗,E∗ = {(e, e ) ∈ E × E : e e }, where e e denotes a directed path with the first and last edges being e and e0 respectively, µ(e) = µ(e0) = ? and any other edge e00 in-between (if any) has constant weight 00 Fig. 1. Cases 1-3 denote the different atomic connections, whereas in over time, i.e., µ(e ) = 1; Case 4 we can see a composed structure whose interconnections consist 6: ∗ ∗ of directed paths with constant weight edges and static nodes, i.e., without EU (E),E = {(uj, e): e ∈ Ej}; self-loops. 7: ∗ ∗ EU (N),E = {(u|J |+i, e ≡ (k, l)) : e ∈ E, k ∈ 0 Ni, µ(e) = ?} ∪ {(u|J |+i, e ): e ≡ (k, l) ∈ E, k ∈ 0 0 In addition, it readily follows that the computation com- Ni, µ(e) = 1, e 1 e }, where e 1 e denotes a directed path with the first and last edges being e and plexity of the procedure presented in Algorithm 1 is as e0 respectively, with µ(e0) = ? and any other edge e00 follows. has constant weight over time, i.e., µ(e00) = 1; Theorem 2: Let (D = (V, EV,V ), w, µ, E = E N {Ej}j∈J , N = {Ni}i∈I , {cj }j∈J , {ci }i∈I ) be a dynamic-flow network. Then, Algorithm 1 has polynomial computational complexity given by O(|V| + |EV,V |).  edges as well as a self-loop that do not have constant Proof: Step 5 and 7 can be implemented by considering weight over time, see case 1 in Figure 1. Then, each of a variation of depth-first search [19]. More precisely, to 0 | | the of the outgoing edges have their dynamics given in verify if e 1 e , consider the digraph D = (V, EV,V ) terms of a linear combination of the incoming edges, i.e., where the direction of the edges is reversed with respect + P − 0 e˙l = wl0e0 + m=1,...,p0 wlmem for l = 1, . . . , p and to D, and perform a depth-first search rooted in e where P − e˙0 = w00e0 + m=1,...,p0 w0mem. If the weight of a self- adjacency is considered only if the edge has constant weight. 0 loop in a vertex i does not vary over time then it is the same If e belongs to such tree, then e 1 e . Similar procedure 0 as having a vertex without edges from a dynamic point of applies to e e , where after the search has exhausted the view, see case 2 in Figure 1. An extension of this scenario is constant weight edges, it considers all remaining vertices depicted in case 3 in Figure 1, where two vertices i and j are not in the tree to which a time varying edges connects to. connected by an edge whose weight does not vary over time. All remaining steps have constant computational complexity; In this case, we can understand the total incoming variation thus, the result follows. of the edge dynamics matching the outgoing variation, hence, Next, we show that the flow-dynamic graphs considered the total variation due the incoming edges in vertex j reflects in this paper can lead to an arbitrary meta digraph, once the into the outgoing edges’ dynamics of vertex i. More pre- procedure in Algorithm 1 is used. + 0 0 P − ∗ cisely, we obtain e˙l = wl0e0 + wl0e0 + m=1,...,p0 wlmem Theorem 3: The meta digraph D can be an arbitrary 0 0 P − for l = 1, . . . , p, e˙0 = w00e0 + wl0e0 + m=1,...,p0 w0mem digraph.  0 0 0 P − ∗ and e˙0 = wl0e0 + m=1,...,p0 w0mem. Further, notice that Proof: Suppose that we want to obtain D = any graph can be recursively created by a sequence of the (X , EX ,X ), then we just need to consider D(E¯) = previous cases; in particular, if we connect two nodes i (W, EW,W ) such that W = X ∪ {e1, . . . , e|EX ,X |}, where and j with outgoing and incoming edges with time varying µ(ei) = 1 and µ(xj) = ? for all i and j, respectively. In weights, respectively, through a directed path containing only addition, let E = {ex, . . . , ex }, and set e to be an X ,X 1 |EX ,X | i x static nodes and constant weight edges, then it is the same as outgoing edge of xj and incoming edge in xk for ei = having the different incoming edges in node j contributing (xk, xj). Then, by executing Algorithm 1 when (D(E¯), µ)

5761 is considered, we obtain D∗. Figure 2 depicts an illustrative tional connectivity from epileptic human brain. Furthermore, example of the proposed methodology. we demonstrate how structural controllability of functional networks can be tested to ensure efficacy of implantable device that target specific brain regions for drug-resistant epilepsy.

V. A CASE STUDY:EPILEPSY-ACTUATION SCHEMES In this section, we ask how structural controllability can be assessed for therapies that actuate neural pathways to treat brain diseases. For instance, assessing structural controllability of epileptic networks is vital for predicting Fig. 2. In this figure, we depict the reduction using Algorithm 1 from whether implantable devices will be effective at controlling the dynamic-flow network in (a) to the meta digraph in (b). In particular, the obtained graph is a 4-vertices star graph which is one of the forbidden seizures. To demonstrate such an application, we pursue the subgraphs in a line graph [20], [21]. case-study of a 20 year-old, male undergoing surgical treat- ment for drug-resistant epilepsy believed to be neocortical Remark 1: Theorem 3 is relevant to contrast our work origin. As a part of routine clinical workup, the patient with that in [17] where the analysis is done through the line underwent implantation of intracranial, subdural electrodes graph L(G), i.e., a graph where the edges of the original for continuous monitoring of the electrocorticogram (ECoG) graph G are the vertices of the line graph L(G) and an edge to localize regions of the epileptic network responsible for between two vertices exist in L(G) if two edges in G are generating seizures. De-identified patient data was retrieved incident in the same vertex. In particular, we notice that the from the online International Epilepsy Electrophysiology line graphs cannot be an arbitrary graph; more specifically, Portal (IEEG Portal) [23]. To demonstrate our structural they cannot contain nine forbidden subgraphs [20], [21]; controllability technique, we analyzed data from a patient in fact, an instance of such forbidden graphs is the 4-star with 84 intracranially, implanted electrodes and one 107 network depicted in Figure 2-(b).  second long seizure (as defined by routine clinical marking). Next, due to Theorem 3, we can obtain the corollaries As a control, we also pulled the 107 second interval leading stated next. to seizure-onset (referred to as pre-seizure). Corollary 1: Determining the minimum cost in-edge in- To test the efficacy of actuating different network re- puts can be achieved in O(nω), where n is the number of gions for seizure control, we first estimated the time-varying state variables in the meta digraph, and ω < 2.373 is the functional connectivity of the epileptic network during the lowest exponent known associated with the complexity of pre-seizure period. To measure functional connectivity, we multiplying two n × n matrices.  divided the pre-seizure ECoG signal into 1-second, non- Proof: It follows from [10], where a polynomial algo- overlapping, wide-sense stationary time windows and applied rithm, with the aferomentioned computational complexity, to a normalized cross-correlation similarity function between determine the minimum cost of dedicated inputs that ensure every possible pair of signals. This procedure yields T sym- structural controllability is proposed. metric, weighted N × N time-varying adjacency matrices, Corollary 2: Determining the minimum cost multi-edge where N is the number of nodes (ECoG channels) and T is and out-node inputs is NP-hard for general dynamic-flow the number of time windows. For our example seizure and networks.  pre-seizure data, T = 107 and N = 84. Proof: The complexity of using multi-edge inputs Next, we asked which of the strongest functional connec- follows from [12], whereas the complexity of using out-node tions are time-varying or most time-constant during the pre- inputs, i.e., multi-edges inputs that conform with the digraph seizure period. For each unique edge, we measured the aver- topology, follows from [22]. age edge strength (Fig. 3) and coefficient of variation (Fig. 4), Although both the minimum cost multi-edge and out-node i.e., the ratio of standard deviation to mean of edge strength, inputs problem are NP-hard in general, they are polynomially over all T time windows during the pre-seizure period. solvable for a large class of problems, as we obtain in the We retained the 25% strongest edges, of which half the next result. edges with lowest coefficient of variation were considered Corollary 3: If the states of the meta digraph belong to time-constant and with highest coefficient of variation con- a single strongly connected component, then the minimum sidered time-varying, see Fig. 5. The resulting adjacency cost multi-edge and out-node inputs is polynomially solvable matrix representing time-constant and time-varying edges is in O(n3).  demonstrated in Fig. 6. Proof: It follows from [13], where a polynomial Using Algorithm 1, we generate the meta digraph, which algorithm to determine the minimum cost multi-edge and describes the dynamics of a given edge in terms of all out-node inputs was proposed for strongly connected state other interacting edges (Fig. 7), and the control input matrix, digraphs. which describes the influence of the actuating nodes, in In the next section, we provide an application of the this case chosen as seizure-onset nodes. The control input present framework in terms of modeling time-varying func- matrix specifies which state variables in the meta digraph,

5762 Average Pre−Seizure Edge Strength Thresholding for Constant and Variable Edges 0.8 0.5

10 0.7 20 0.4 0.6 30 0.5 0.3 40 0.4 50 0.2 0.3

Network Nodes 60

0.2 Coefficient of Variation 70 0.1 0.1 80 0 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 Network Nodes Average Edge Strength

Fig. 3. Average edge strength in dynamic-flow network from 107 time Fig. 5. Relationship between edge strength and coefficient of variation windows preceding seizure onset. Edges strength is the statistical similarity for all edges in the dynamic-flow network. Vertical red line demarcates between all possible node (electrode) pairs in the network. Colors represent the threshold used to retain 25% strongest edges. Horizontal blue line is the average edge strength. Clinically-defined seizure-onset nodes are 29 and 37. threshold by which edges with lower coefficient of variation were considered time-constant and greater coefficient of variation were considered time- varying. Pre−Seizure Edge Coefficient of Variation

Pre-Seizure Constant and Variable Edges 10 0.45 2

20 0.4 10

30 0.35 20 1.5 40 0.3 30

50 0.25 40 1

Network Nodes 60 0.2 50

70 0.15 Network Nodes 60 0.5 80 0.1 70

20 40 60 80 80 Network Nodes 0 20 40 60 80 Network Nodes Fig. 4. Coefficient of variation of edge strength in dynamic-flow network from 107 time windows preceding seizure onset. Colors represent the stability of edge strength over time windows. Fig. 6. Adjacency matrix demonstrated edge stability (green) and variability (red). i.e., the edges in the dynamic-flow network, are actuated. For a point of comparison, we construct the meta digraph digraph, we find that any single in-edge input within the under two conditions: (i) real formulation of observed time- seizure-onset zone guarantees structural controllability in the constant and time-varying edges; (ii) extreme case where epileptic network for the given patient. These results support all the strongest edges are considered time-varying, i.e., no the capability of current actuating technologies under the time-constant edges exist , as depicted in Fig. 8. We find assumption that electrode placement captures the full extent that the original dynamic-flow network is 4.3% sparse, while of the epileptic region. If we assign different cost to actuating the extreme case with strongest edges all considered time- different regions in the epileptic network, due to the dense varying is 98.3% sparse. These findings suggest that in the connectivity (as presented in Fig. 7) any single actuated original dynamic-flow, containing some time-constant edges, region will suffice to ensure structural controllability; thus, has greater chance of reaching an edge from any other given the less costly, or harmless region should be selected. edge. VI.CONCLUSIONSANDFURTHERRESEARCH We ask if actuating nodes in the clinically-defined seizure-onset zone ensures structural controllability of the In this paper, we have extended the notion of generic con- epileptic network. We assume a constant cost of 1 for each trollability for dynamic-flow networks, with either constant in-edge input in the dynamic-flow network, since the regions and free parameters over time, and provided necessary and of interest have similar properties. Using the algorithm sufficient conditions for this to hold. In addition, we address implementing Corollary 1 (see [10] for details) with the meta the problem of minimum actuator placement incurring in

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